Dilute Suspension Viscosity

Apart from chemical composition, an important variable in the description of emulsions is the volume fraction, outer phase. For spherical droplets, of radius a, the volume fraction is given by the number density, n, times the spherical volume, 0 = Ava nl2>. It is easy to show that the maximum packing fraction of spheres is 0 = 0.74 (see Problem XIV-2). Many physical properties of emulsions can be characterized by their volume fraction. The viscosity of a dilute suspension of rigid spheres is an example where the Einstein limiting law is [2]... 

As shown in Figure 2, adsorption of dispersants on particle surfaces can increase 2eta potential further, enhancing electrostatic repulsion. Increased repulsion between particles is evidenced by lower viscosity in concentrated slurries, or decreased settling rates in dilute suspensions. The effect of added dispersants on settling of (anhydrous) iron oxide particles is shown in Figure 3. 

At least, in absolute majority of cases, where the concentration dependence of viscosity is discussed, the case at hand is a shear flow. At the same time, it is by no means obvious (to be more exact the reverse is valid) that the values of the viscosity of dispersions determined during shear, will correlate with the values of the viscosity measured at other types of stressed state, for example at extension. Then a concept on the viscosity of suspensions (except ultimately diluted) loses its unambiguousness, and correspondingly the coefficients cn cease to be characteristics of the system, because they become dependent on the type of flow. 

This is the same as the expression for the viscosity of dilute suspensions (see for example Chapter 3). The expression is however limited to low concentrations of particles, i.e. for tp < 0.05 there are others that can be used throughout the range of volume fractions. Landel26 proposed that the elastic modulus can be described by... 

This value of kn is actually low by an order of magnitude for dilute suspensions of charged spheres of radius Rg. This is due to the neglect of interchain correlations for c < c in the structure factor used in the derivation of Eqs. (295)-(298). If the repulsive interaction between polyelectrolyte chains dominates, as expected in salt-free solutions, the virial expansion for viscosity may be valid over considerable range of concentrations where the average distance between chains scales as. This virial series may be approxi-... 

With this background of non-Newtonian behavior in hand, let us examine the viscous behavior of suspensions and slurries in ceramic systems. For dilute suspensions on noninteracting spheres in a Newtonian liquid, the viscosity of the suspension, r)s, is greater than the viscosity of the pure liquid medium, rjp. In such cases, a relative viscosity, rjr, is utilized, which is defined as rjs/rjL. For laminar flow, is given by the Einstein equation... 

Since n is less than unity, the apparent viscosity decreases with the deformation rate. Examples of such materials are some polymeric solutions or melts such as rubbers, cellulose acetate and napalm suspensions such as paints, mayonnaise, paper pulp, or detergent slurries and dilute suspensions of inert solids. Pseudoplastic properties of wallpaper paste account for good spreading and adhesion, and those of printing inks prevent their running at low speeds yet allow them to spread easily in high speed machines. 

To calculate the characteristics of viscoelasticity in the framework of mesoscopic approach, one can start with the system of entangled macromolecules, considered as a dilute suspension of chains with internal viscoelasticity moving in viscoelastic medium, while the elastic and internal viscosity forces, according to equations (3.4)-(3.6) and (3.8), have the form... 

Here, the sphere center is instantaneously situated at point 0 the sphere center translates with velocity U, while it rotates with angular velocity (a r is measured relative to 0 its magnitude r is denoted by r. Moreover, f = r/r is a unit radial vector. The latter solution is derivable in a variety of ways e.g., from Lamb s (1932) general solution (Brenner, 1970). [Equation (2.12) represents a superposition (Brenner, 1958) of three physically distinct solutions, corresponding, respectively, to (i) translation of a sphere through a fluid at rest at infinity (ii) rotation of a sphere in a fluid at rest at infinity (iii) motion of a neutrally buoyant sphere suspended in a linear shear flow. The latter was first obtained by Einstein (1906, 1911 cf. Einstein, 1956) in connection with his classic calculation of the viscosity of a dilute suspension of spheres, which formed part of his 1905 Ph.D. thesis.]... 

Keeping the particles uniformly distributed throughout the dispersion is an important aspect of physical stability in suspensions. Based on Stokes s law for dilute suspensions where the particles do not interfere with one another, there are different factors that control the velocity of particle sedimentation in a suspension, for instance, particle diameter, densities of the dispersed phase and the dispersion medium, as well as viscosity of the dispersion medium [36]. Remington describes the formulation of trisulfapyrimidines oral suspension [1], In addition, Lieberman et al. [42,48] are also good sources of typical formulations for suspensions. 

The simplest model to predict the viscosity of liquids containing solid particles is that derived for a dilute suspension of uniform, monodisperse, noninteracting hard spheres in a solvent of viscosity rj. ... 

A dilute ceramic suspension has Newtonian rheoU. Thus an important characteristic of a dilute suspension is its viscosity. The viscosity of a dilute suspension, rj, is alwa5re higher than that of the pure solvent, Tjj. Using pure hydrodynamics Einstein [10,11] derived an expression relating the viscosity to the volume fraction, , of the dispersed phase ... 

FIGURE 12S Schematic of the low shear viscosity of TiOj as a function of pH. Near the zero point of charge (ZPC) the rheology is non-Newtonian for dilute suspensions, conforming to the Cross equation, which suggests that aggregation is responsible for this increase in viscosity. Away from the ZPC, the rheology is Newtonian for dilute suspensions. 

Consider a dilute suspension of Np spherical soft particles moving with a velocity U exp(—/fflf) in a symmetrical electrolyte solution of viscosity r] and relative permittivity r in an applied oscillating pressure gradient field Vp exp(—imt) due to a sound wave propagating in the suspension, where m is the angular frequency 2n times frequency) and t is time. We treat the case in which m is low such that the dispersion of r can be neglected. We assume that the particle core of radius a is coated... 

The effective viscosity rj of cl dilute suspension of uncharged coUoidal particles in a liquid is greater than the viscosity t] of the original liquid. Einstein [1] derived the following expression for... 

In this chapter, we first present a theory of the primary electroviscous effect in a dilute suspension of soft particles, that is, particles covered with an ion-penetrable surface layer of charged or uncharged polymers. We derive expressions for the effective viscosity and the primary electroviscous coefficient of a dilute suspension of soft particles [26]. We then derive an expression for the effective viscosity of uncharged porous spheres (i.e., spherical soft particles with no particle core) [27]. 

Einstein s equation for the viscosity of a dilute suspension of spherical particles is... 

Figure 6.14 Intrinsic viscosity versus Peclet numberfor dilute suspensions of spheroidal particles of (a) oblate shape and (b) prolate shape, (From Macosko 1994, adapted from Brenner 1974, with permission from Pergamon Press.)...

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